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A328599
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Number of compositions of n with no part circularly followed by a divisor or a multiple.
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8
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1, 0, 0, 0, 0, 2, 0, 4, 2, 4, 12, 8, 22, 14, 36, 44, 62, 114, 130, 206, 264, 414, 602, 822, 1250, 1672, 2520, 3518, 5146, 7408, 10448, 15224, 21496, 31284, 44718, 64170, 92314, 131618, 190084, 271870, 391188, 560978, 804264, 1155976, 1656428, 2381306, 3414846
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OFFSET
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0,6
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COMMENTS
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A composition of n is a finite sequence of positive integers summing to n.
Circularity means the last part is followed by the first.
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LINKS
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EXAMPLE
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The a(0) = 1 through a(12) = 22 compositions (empty columns not shown):
() (2,3) (2,5) (3,5) (2,7) (3,7) (2,9) (5,7)
(3,2) (3,4) (5,3) (4,5) (4,6) (3,8) (7,5)
(4,3) (5,4) (6,4) (4,7) (2,3,7)
(5,2) (7,2) (7,3) (5,6) (2,7,3)
(2,3,5) (6,5) (3,2,7)
(2,5,3) (7,4) (3,4,5)
(3,2,5) (8,3) (3,5,4)
(3,5,2) (9,2) (3,7,2)
(5,2,3) (4,3,5)
(5,3,2) (4,5,3)
(2,3,2,3) (5,3,4)
(3,2,3,2) (5,4,3)
(7,2,3)
(7,3,2)
(2,3,2,5)
(2,3,4,3)
(2,5,2,3)
(3,2,3,4)
(3,2,5,2)
(3,4,3,2)
(4,3,2,3)
(5,2,3,2)
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MATHEMATICA
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Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], And@@Not/@Divisible@@@Partition[#, 2, 1, 1]&&And@@Not/@Divisible@@@Reverse/@Partition[#, 2, 1, 1]&]], {n, 0, 10}]
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PROG
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(PARI)
b(n, q, pred)={my(M=matrix(n, n)); for(k=1, n, M[k, k]=pred(q, k); for(i=1, k-1, M[i, k]=sum(j=1, k-i, if(pred(j, i), M[j, k-i], 0)))); M[q, ]}
seq(n)={concat([1], sum(k=1, n, b(n, k, (i, j)->i%j<>0&&j%i<>0)))} \\ Andrew Howroyd, Oct 26 2019
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CROSSREFS
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The case forbidding only divisors (not multiples) is A328598.
The non-circular version is A328508.
Partitions with no part followed by a divisor are A328171.
Cf. A000740, A008965, A167606, A318729, A318748, A328460, A328593, A328600, A328603, A328608, A328609, A328674.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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